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\title{An explicit power saving for the exact discrete Fourier transform}
\author{OpenAI}
\date{September 25, 2026}
\begin{document}
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\begin{abstract}
We give a deterministic algorithm that computes the discrete Fourier transform at every length $n$ in $O(n(\log n)^{1-10^{-13}})$ operations. The model uses exact complex arithmetic, unrestricted coefficients, specified Fourier roots, and unit-cost logarithmic-size indexing; scalar preparation and array organization are included.
\end{abstract}
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